Group #13105078 — subgroups

Subgroups — #13105078

Order 1600.

552 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₅×#4016	C₂	1	2	yes	maximal, complemented
3	#11717037	C₂	1	2	yes	maximal, complemented
4	#11716984	C₂	1	4	yes	maximal
5	#11716967	C₂	1	2	yes	maximal
6	#11716956	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₅⋊D₅	C₂²	1	2	yes	—
8	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
9	C₂×#6039	C₂²	1	4	yes	—
10	C₄²×C₅²	C₂²	1	1	yes	characteristic
11	C₄×#1039	C₂²	1	2	yes	—
12	C₄×#1039	C₂²	1	1	yes	characteristic
13	C₅×C₄⋊Dic₅	C₂²	1	4	yes	—
14	C₅×#683	C₂²	1	8	yes	—
15	#87781	C₂²	1	4	yes	—
16	#87779	C₂²	1	4	yes	—
17	Dic₅²	C₂²	1	4	yes	—
18	#66022	—	5	2	no	maximal
19	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₅²	C₂³	1	2	yes	—
21	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
22	C₂×#1039	C₂³	1	1	yes	characteristic
23	C₂×#1039	C₂³	1	2	yes	—
24	C₄×C₅⋊D₅	—	2	4	no	—
25	C₄×C₅⋊D₅	Q₈	1	4	yes	complemented
26	C₂×C₅×Dic₅	C₂³	1	8	yes	—
27	#6039	—	2	8	no	—
28	C₅×#125	—	5	2	no	—
29	#4042	—	5	8	no	—
30	#4038	—	5	4	no	—
31	#4034	—	5	4	no	—
32	#4033	—	5	8	no	—
33	C₄²×D₅	—	5	2	no	—
34	C₄²×D₅	—	10	2	no	—
35	#4016	D₅	1	2	yes	complemented
36	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
37	C₂×C₅⋊D₅	—	2	2	no	—
38	C₂×C₅⋊D₅	C₂×Q₈	1	2	yes	—
39	C₄×C₅²	—	2	2	no	—
40	C₄×C₅²	C₂×Q₈	1	2	yes	—
41	#1039	C₂×Q₈	1	2	yes	—
42	#1039	#41	1	4	yes	—
43	C₅×Dic₅	—	2	8	no	—
44	C₂×C₄×D₅	—	5	2	no	—
45	C₂×C₄×D₅	—	5	4	no	—
46	C₂×C₄×D₅	—	10	2	no	—
47	C₂×C₄×D₅	—	10	4	no	—
48	C₂×Dic₁₀	—	5	8	no	—
49	C₅×C₄⋊C₄	—	5	8	no	—
50	C₅×C₄⋊C₄	—	5	4	no	—
51	C₄²×C₅	—	2	2	no	—
52	C₄²×C₅	C₂×D₅	1	2	yes	—
53	#685	—	5	8	no	—
54	C₄⋊Dic₅	C₂×D₅	1	4	yes	—
55	#683	—	5	8	no	—
56	#683	C₂×D₅	1	8	yes	—
57	C₄×Dic₅	—	5	4	no	—
58	C₄×Dic₅	—	5	2	no	—
59	C₄×Dic₅	—	5	8	no	—
60	C₄×Dic₅	—	10	4	no	—
61	C₄×Dic₅	—	10	2	no	—
62	#533	—	25	1	no	Sylow
63	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₅²	C₂×#41	1	2	yes	—
65	C₅⋊D₅	—	2	2	no	—
66	C₂²×D₅	—	5	2	no	—
67	C₂²×D₅	—	10	2	no	—
68	C₂×C₄×C₅	—	2	4	no	—
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	—	5	8	no	—
71	C₂×C₄×C₅	C₂²×D₅	1	4	yes	—
72	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
73	C₂×Dic₅	—	5	2	no	—
74	C₂×Dic₅	—	5	4	no	—
75	C₂×Dic₅	—	10	2	no	—
76	C₂×Dic₅	—	10	4	no	—
77	C₂×Dic₅	C₂²×D₅	1	8	yes	—
78	C₄×D₅	—	5	8	no	—
79	C₄×D₅	—	10	8	no	—
80	C₄×D₅	—	10	8	no	—
81	C₄×D₅	—	10	16	no	—
82	Dic₁₀	—	10	16	no	—
83	C₄⋊Q₈	—	25	2	no	—
84	#125	—	5	2	no	—
85	C₂²⋊Q₈	—	25	4	no	—
86	C₄×Q₈	—	25	4	no	—
87	#117	—	25	2	no	—
88	C₂×C₄²	—	25	1	no	—
89	C₅²	#533	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₅	—	2	2	no	—
91	C₂²×C₅	C₂³×D₅	1	2	yes	—
92	C₂×D₅	—	5	4	no	—
93	C₂×D₅	—	10	4	no	—
94	C₂×D₅	—	10	8	no	—
95	C₂×D₅	—	10	4	no	—
96	C₄×C₅	—	2	8	no	—
97	C₄×C₅	—	2	4	no	—
98	C₄×C₅	—	2	4	no	—
99	C₄×C₅	—	10	8	no	—
100	C₄×C₅	Q₈×D₅	1	4	yes	—
101	Dic₅	—	2	8	no	—
102	Dic₅	—	5	4	no	—
103	Dic₅	—	5	8	no	—
104	Dic₅	—	10	4	no	—
105	Dic₅	—	10	8	no	—
106	C₂×Q₈	—	25	4	no	—
107	C₂²×C₄	—	25	1	no	—
108	C₂²×C₄	—	25	2	no	—
109	C₄⋊C₄	—	5	4	no	—
110	C₄⋊C₄	—	5	8	no	—
111	C₄⋊C₄	—	25	4	no	—
112	C₂²⋊C₄	—	25	4	no	—
113	C₄²	—	25	4	no	—
114	C₄²	—	25	2	no	—
115	C₄²	—	25	1	no	—
116	C₄²	D₅²	1	1	yes	characteristic
117	C₂×C₅	—	2	2	no	—
118	C₂×C₅	—	2	4	no	—
119	C₂×C₅	C₂×Q₈×D₅	1	2	yes	—
120	C₂×C₅	D₅×#41	1	4	yes	—
121	D₅	—	10	8	no	—
122	D₅	—	10	4	no	—
123	C₂³	—	25	1	no	—
124	Q₈	—	50	8	no	—
125	C₂×C₄	—	5	8	no	—
126	C₂×C₄	—	25	1	no	—
127	C₂×C₄	—	25	2	no	—
128	C₂×C₄	—	25	4	no	—
129	C₂×C₄	—	50	4	no	—
130	C₂×C₄	C₂×D₅²	1	2	yes	—
131	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
132	C₅	—	2	2	no	—
133	C₅	#66022	1	2	yes	complemented
134	C₂²	—	25	2	no	—
135	C₂²	—	50	2	no	—
136	C₂²	C₂²×D₅²	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	10	8	no	—
139	C₄	—	25	2	no	—
140	C₄	—	25	4	no	—
141	C₄	#87872	1	2	yes	—
142	C₂	—	50	2	no	—
143	C₂	C₂×#87872	1	1	yes	characteristic
144	C₂	#11717471	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×C₅⋊D₅

C₂

yes

maximal, characteristic

C₅×#4016

C₂

yes

maximal, complemented

#11717037

C₂

yes

maximal, complemented

#11716984

C₂

yes

maximal

#11716967

C₂

yes

maximal

#11716956

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

—

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂×#6039

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

C₄×#1039

C₂²

yes

—

C₄×#1039

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₅×#683

C₂²

yes

—

#87781

C₂²

yes

—

#87779

C₂²

yes

—

Dic₅²

C₂²

yes

—

#66022

—

maximal

C₂²×C₅⋊D₅

C₂³

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

C₂³

yes

characteristic

C₂×#1039

C₂³

yes

—

C₄×C₅⋊D₅

—

C₄×C₅⋊D₅

Q₈

yes

complemented

C₂×C₅×Dic₅

C₂³

yes

—

—

—

—

—

—

—

C₄²×D₅

—

C₄²×D₅

—

#4016

D₅

yes

complemented

C₂²×C₅²

C₂⁴

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₂×Q₈

yes

—

C₄×C₅²

—

C₄×C₅²

C₂×Q₈

yes

—

#1039

C₂×Q₈

yes

—

#1039

#41

yes

—

C₅×Dic₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂×D₅

yes

—

#685

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

#683

C₂×D₅

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

#533

—

Sylow

C₂×C₅²

C₂²×Q₈

yes

characteristic

C₂×C₅²

C₂×#41

yes

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂²×D₅

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₅²

#533

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂³×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

100

C₄×C₅

Q₈×D₅

yes

—

101

Dic₅

—

102

Dic₅

—

103

Dic₅

—

104

Dic₅

—

105

Dic₅

—

106

C₂×Q₈

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

D₅²

yes

characteristic

117

C₂×C₅

—

118

C₂×C₅

—

119

C₂×C₅

C₂×Q₈×D₅

yes

—

120

C₂×C₅

D₅×#41

yes

—

121

D₅

—

122

D₅

—

123

C₂³

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×D₅²

yes

—

131

C₂×C₄

C₂×D₅²

yes

characteristic

132

C₅

—

133

C₅

#66022

yes

complemented

134

C₂²

—

135

C₂²

—

136

C₂²

C₂²×D₅²

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#87872

yes

—

142

C₂

—

143

C₂

C₂×#87872

yes

characteristic

144

C₂

#11717471

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₄²×C₅²	C₂²
prime radical	C₂²×C₅⋊D₅	C₂³
hypercenter	C₄²	D₅²
hypocenter	C₅²	#533
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center, Frattini

C₂²

C₂²×D₅²

commutator, socle, cosocle

C₂²×C₅²

C₂⁴

Fitting

C₄²×C₅²

C₂²

prime radical

C₂²×C₅⋊D₅

C₂³

hypercenter

C₄²

D₅²

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
2. C₅×#4016	C₂	semidirect
3. #11717037	C₂	semidirect
6. #11716956	C₂	semidirect
25. C₄×C₅⋊D₅	Q₈	semidirect
35. #4016	D₅	semidirect
89. C₅²	#533	semidirect
133. C₅	#66022	semidirect

H ≅ G/N

type

2. C₅×#4016

C₂

semidirect

3. #11717037

C₂

semidirect

6. #11716956

C₂

semidirect

25. C₄×C₅⋊D₅

Q₈

semidirect

35. #4016

D₅

semidirect

89. C₅²

#533

semidirect

133. C₅

#66022

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄²×C₅⋊D₅	2
C₅×#4016	2
#11717037	2
#11716984	2
#11716967	2
#11716956	2
#66022	5

subgroup

index

C₄²×C₅⋊D₅

Subgroups — #13105078

Conjugacy classes of subgroups

Complements

Maximal subgroups